Optimal. Leaf size=65 \[ \frac{5 x^2}{2}-\frac{25 \left (x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}-\frac{17}{4} \log \left (x^4+2 x^2+3\right )-\frac{17 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
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Rubi [A] time = 0.105306, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {1663, 1660, 1657, 634, 618, 204, 628} \[ \frac{5 x^2}{2}-\frac{25 \left (x^2+3\right )}{8 \left (x^4+2 x^2+3\right )}-\frac{17}{4} \log \left (x^4+2 x^2+3\right )-\frac{17 \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )}{8 \sqrt{2}} \]
Antiderivative was successfully verified.
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Rule 1663
Rule 1660
Rule 1657
Rule 634
Rule 618
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3 \left (4+x^2+3 x^4+5 x^6\right )}{\left (3+2 x^2+x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x \left (4+x+3 x^2+5 x^3\right )}{\left (3+2 x+x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac{25 \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \frac{-50-56 x+40 x^2}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=-\frac{25 \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}+\frac{1}{16} \operatorname{Subst}\left (\int \left (40-\frac{34 (5+4 x)}{3+2 x+x^2}\right ) \, dx,x,x^2\right )\\ &=\frac{5 x^2}{2}-\frac{25 \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{17}{8} \operatorname{Subst}\left (\int \frac{5+4 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac{5 x^2}{2}-\frac{25 \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{17}{8} \operatorname{Subst}\left (\int \frac{1}{3+2 x+x^2} \, dx,x,x^2\right )-\frac{17}{4} \operatorname{Subst}\left (\int \frac{2+2 x}{3+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac{5 x^2}{2}-\frac{25 \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{17}{4} \log \left (3+2 x^2+x^4\right )+\frac{17}{4} \operatorname{Subst}\left (\int \frac{1}{-8-x^2} \, dx,x,2 \left (1+x^2\right )\right )\\ &=\frac{5 x^2}{2}-\frac{25 \left (3+x^2\right )}{8 \left (3+2 x^2+x^4\right )}-\frac{17 \tan ^{-1}\left (\frac{1+x^2}{\sqrt{2}}\right )}{8 \sqrt{2}}-\frac{17}{4} \log \left (3+2 x^2+x^4\right )\\ \end{align*}
Mathematica [A] time = 0.0271538, size = 61, normalized size = 0.94 \[ \frac{1}{16} \left (40 x^2-\frac{50 \left (x^2+3\right )}{x^4+2 x^2+3}-68 \log \left (x^4+2 x^2+3\right )-17 \sqrt{2} \tan ^{-1}\left (\frac{x^2+1}{\sqrt{2}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 59, normalized size = 0.9 \begin{align*}{\frac{5\,{x}^{2}}{2}}-{\frac{1}{2\,{x}^{4}+4\,{x}^{2}+6} \left ({\frac{25\,{x}^{2}}{4}}+{\frac{75}{4}} \right ) }-{\frac{17\,\ln \left ({x}^{4}+2\,{x}^{2}+3 \right ) }{4}}-{\frac{17\,\sqrt{2}}{16}\arctan \left ({\frac{ \left ( 2\,{x}^{2}+2 \right ) \sqrt{2}}{4}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47108, size = 73, normalized size = 1.12 \begin{align*} \frac{5}{2} \, x^{2} - \frac{17}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{25 \,{\left (x^{2} + 3\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{17}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50264, size = 219, normalized size = 3.37 \begin{align*} \frac{40 \, x^{6} + 80 \, x^{4} - 17 \, \sqrt{2}{\left (x^{4} + 2 \, x^{2} + 3\right )} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) + 70 \, x^{2} - 68 \,{\left (x^{4} + 2 \, x^{2} + 3\right )} \log \left (x^{4} + 2 \, x^{2} + 3\right ) - 150}{16 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.167923, size = 66, normalized size = 1.02 \begin{align*} \frac{5 x^{2}}{2} - \frac{25 x^{2} + 75}{8 x^{4} + 16 x^{2} + 24} - \frac{17 \log{\left (x^{4} + 2 x^{2} + 3 \right )}}{4} - \frac{17 \sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} x^{2}}{2} + \frac{\sqrt{2}}{2} \right )}}{16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1123, size = 73, normalized size = 1.12 \begin{align*} \frac{5}{2} \, x^{2} - \frac{17}{16} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (x^{2} + 1\right )}\right ) - \frac{25 \,{\left (x^{2} + 3\right )}}{8 \,{\left (x^{4} + 2 \, x^{2} + 3\right )}} - \frac{17}{4} \, \log \left (x^{4} + 2 \, x^{2} + 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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